A History of Mathematics/Antiquity/The Egyptians
THE EGYPTIANS.
Though there is great difference of opinion regarding the antiquity of Egyptian civilisation, yet all authorities agree in the statement that, however far back they go, they find no uncivilised state of society. "Menes, the first king, changes the course of the Nile, makes a great reservoir, and builds the temple of Phthah at Memphis." The Egyptians built the pyramids at a very early period. Surely a people engaging in enterprises of such magnitude must have known something of mathematics—at least of practical mathematics.
All Greek writers are unanimous in ascribing, without envy, to Egypt the priority of invention in the mathematical sciences. Plato in Phœdrus says: "At the Egyptian city of Naucratis there was a famous old god whose name was Theuth; the bird which is called the Ibis was sacred to him, and he was the inventor of many arts, such as arithmetic and calculation and geometry and astronomy and draughts and dice, but his great discovery was the use of letters."
Aristotle says that mathematics had its birth in Egypt, because there the priestly class had the leisure needful for the study of it. Geometry, in particular, is said by Herodotus, Diodorus, Diogenes Laertius, Iamblichus, and other ancient writers to have originated in Egypt.[5] In Herodotus we find this (II. c. 109): "They said also that this king [Sesostris] divided the land among all Egyptians so as to give each one a quadrangle of equal size and to draw from each his revenues, by imposing a tax to be levied yearly. But every one from whose part the river tore away anything, had to go to him and notify what had happened; he then sent the overseers, who had to measure out by how much the land had become smaller, in order that the owner might pay on what was left, in proportion to the entire tax imposed. In this way, it appears to me, geometry originated, which passed thence to Hellas."
We abstain from introducing additional Greek opinion regarding Egyptian mathematics, or from indulging in wild conjectures. We rest our account on documentary evidence. A hieratic papyrus, included in the Rhind collection of the British Museum, was deciphered by Eisenlohr in 1877, and found to be a mathematical manual containing problems in arithmetic and geometry. It was written by Ahmes some time before 1700 B.C., and was founded on an older work believed by Birch to date back as far as 3400 B.C.! This curious papyrus—the most ancient mathematical handbook known to us—puts us at once in contact with the mathematical thought in Egypt of three or five thousand years ago. It is entitled "Directions for obtaining the Knowledge of all Dark Things." We see from it that the Egyptians cared but little for theoretical results. Theorems are not found in it at all. It contains "hardly any general rules of procedure, but chiefly mere statements of results intended possibly to be explained by a teacher to his pupils."[6] In geometry the forte of the Egyptians lay in making constructions and determining areas. The area of an isosceles triangle, of which the sides measure 10 ruths and the base 4 ruths, was erroneously given as 20 square ruths, or half the product of the base by one side. The area of an isosceles trapezoid is found, similarly, by multiplying half the sum of the parallel sides by one of the non-parallel sides. The area of a circle is found by deducting from the diameter of its length and squaring the remainder. Here is taken , a very fair approximation.[6] The papyrus explains also such problems as these,—To mark out in the field a right triangle whose sides are 10 and 4 units; or a trapezoid whose parallel sides are 6 and 4, and the non-parallel sides each 20 units.
Some problems in this papyrus seem to imply a rudimentary knowledge of proportion.
The base-lines of the pyramids run north and south, and east and west, but probably only the lines running north and south were determined by astronomical observations. This, coupled with the fact that the word harpedonaptæ, applied to Egyptian geometers, means "rope-stretchers," would point to the conclusion that the Egyptian, like the Indian and Chinese geometers, constructed a right triangle upon a given line, by stretching around three pegs a rope consisting of three parts in the ratios 3:4:5, and thus forming a right triangle.[3] If this explanation is correct, then the Egyptians were familiar, 2000 years B.C., with the well-known property of the right triangle, for the special case at least when the sides are in the ratio 3:4:5.
On the walls of the celebrated temple of Horus at Edfu have been found hieroglyphics, written about 100 B.C., which enumerate the pieces of land owned by the priesthood, and give their areas. The area of any quadrilateral, however irregular, is there found by the formula . Thus, for a quadrangle whose opposite sides are 5 and 8, 20 and 15, is given the area .[7] The incorrect formulæ of Ahmes of 3000 years B.C. yield generally closer approximations than those of the Edfu inscriptions, written 200 years after Euclid!
The fact that the geometry of the Egyptians consists chiefly of constructions, goes far to explain certain of its great defects. The Egyptians failed in two essential points without which a science of geometry, in the true sense of the word, cannot exist. In the first place, they failed to construct a rigorously logical system of geometry, resting upon a few axioms and postulates. A great many of their rules, especially those in solid geometry, had probably not been proved at all, but were known to be true merely from observation or as matters of fact. The second great defect was their inability to bring the numerous special cases under a more general view, and thereby to arrive at broader and more fundamental theorems. Some of the simplest geometrical truths were divided into numberless special cases of which each was supposed to require separate treatment.
Some particulars about Egyptian geometry can be mentioned more advantageously in connection with the early Greek mathematicians who came to the Egyptian priests for instruction.
An insight into Egyptian methods of numeration was obtained through the ingenious deciphering of the hieroglyphics by Champollion, Young, and their successors. The symbols used were the following: 
for 1, 
for 10, 
for 100, 
for 1000, 
for 10,000, 
for 100,000, 
for 1,000,000, 
for 10,000,000.[3] The symbol for 1 represents a vertical staff; that for 10,000 a pointing finger; that for 100,000 a burbot; that for 1,000,000, a man in astonishment. The significance of the remaining symbols is very doubtful. The writing of numbers with these hieroglyphics was very cumbrous. The unit symbol of each order was repeated as many times as there were units in that order. The principle employed was the additive. Thus, 23 was written 
Besides the hieroglyphics, Egypt possesses the hieratic and demotic writings, but for want of space we pass them by.
Herodotus makes an important statement concerning the mode of computing among the Egyptians. He says that they "calculate with pebbles by moving the hand from right to left, while the Hellenes move it from left to right." Herein we recognise again that instrumental method of figuring so extensively used by peoples of antiquity. The Egyptians used the decimal scale. Since, in figuring, they moved their hands horizontally, it seems probable that they used ciphering-boards with vertical columns. In each column there must have been not more than nine pebbles, for ten pebbles would be equal to one pebble in the column next to the left.
The Ahmes papyrus contains interesting information on the way in which the Egyptians employed fractions. Their methods of operation were, of course, radically different from ours. Fractions were a subject of very great difficulty with the ancients. Simultaneous changes in both numerator and denominator were usually avoided. In manipulating fractions the Babylonians kept the denominators (60) constant. The Romans likewise kept them constant, but equal to 12. The Egyptians and Greeks, on the other hand, kept the numerators constant, and dealt with variable denominators. Ahmes used the term "fraction" in a restricted sense, for he applied it only to unit-fractions, or fractions having unity for the numerator. It was designated by writing the denominator and then placing over it a dot. Fractional values which could not be expressed by any one unit-fraction were expressed as the sum of two or more of them. Thus, he wrote in place of . The first important problem naturally arising was, how to represent any fractional value as the sum of unit-fractions. This was solved by aid of a table, given in the papyrus, in which all fractions of the form (where n designates successively all the numbers up to 49) are reduced to the sum of unit-fractions. Thus, ; . When, by whom, and how this table was calculated, we do not know. Probably it was compiled empirically at different times, by different persons. It will be seen that by repeated application of this table, a fraction whose numerator exceeds two can be expressed in the desired form, provided that there is a fraction in the table having the same denominator that it has. Take, for example, the problem, to divide 5 by 21. In the first place, . From the table we get . Then . The papyrus contains problems in which it is required that fractions be raised by addition or multiplication to given whole numbers or to other fractions. For example, it is required to increase to 1. The common denominator taken appears to be 45, for the numbers are stated as , , , , 1. The sum of these is forty-fifths. Add to this and the sum is . Add , and we have 1. Hence the quantity to be added to the given fraction is .
Having finished the subject of fractions, Ahmes proceeds to the solution of equations of one unknown quantity. The unknown quantity is called 'hau' or heap. Thus the problem, "heap, its , its whole, it makes 19," i.e. . In this case, the solution is as follows: ; ; . But in other problems, the solutions are effected by various other methods. It thus appears that the beginnings of algebra are as ancient as those of geometry.
The principal defect of Egyptian arithmetic was the lack of a simple, comprehensive symbolism—a defect which not even the Greeks were able to remove.
The Ahmes papyrus doubtless represents the most advanced attainments of the Egyptians in arithmetic and geometry. It is remarkable that they should have reached so great proficiency in mathematics at so remote a period of antiquity. But strange, indeed, is the fact that, during the next two thousand years, they should have made no progress whatsoever in it. The conclusion forces itself upon us, that they resemble the Chinese in the stationary character, not only of their government, but also of their learning. All the knowledge of geometry which they possessed when Greek scholars visited them, six centuries B.C., was doubtless known to them two thousand years earlier, when they built those stupendous and gigantic structures—the pyramids. An explanation for this stagnation of learning has been sought in the fact that their early discoveries in mathematics and medicine had the misfortune of being entered upon their sacred books and that, in after ages, it was considered heretical to augment or modify anything therein. Thus the books themselves closed the gates to progress.